and probit models and that both are better than the linear probability model. From the practical point of view it appears that the squared correlation between D, and Di and Effrons R- are sufficient for many problems.

Since we decided on the probit and logit models and was not significant in these models, we decided to drop that variable and reestimate the probit and logit models. The revised estimates were (figures in parentheses are asymptotic r-ratios)

Logit

Probit

D, = 16.57 -I- 0.0165F + 9.13F - 0.715LF + 85.367VW

(0.84) (1.72) (1.81) (-1.49) (2.38)

¹„ £>,) = 0.5982 Effrons 7? = 0.5982

Cragg-Uhlers R = 0.7077 McFaddens R = 0.5914

D, = 10.27 + 0.0094F + 5.55F - 0.437LF + 50.25NW

(0.98) (1.86) (1.97) (- 1.7) (2.50)

R\Di, bd = 0.5950 Effrons R = 0.5947

Cragg-Uhlers 7? = 0.7113 McFaddens 7? = 0.5955

Again, to make the logit coefficients comparable to the probit coefficients, we have to divide the former by 1.6. This gives 10.36, 0.0103, 5.71, -0.447, and 53.35, respectively, which are close to the probit coefficients.

8.11 Truncated Variables: The Tobit Model

In our discussion of the logit and probit models we talked about a latent variable y* which was not observed, for which we could specify the regression model:

yJ = , + , (8.18)

For simpUcity of exposition we are assuming that there is only one explanatory variable. In the logit and probit models, what we observe is a dummy variable

Table 8.5 Different R- Measures for the Logit, Probit, and Linear Probability Models

Linear

Logit Probit Probability

ify- > 0 ify- 0

yi =

Suppose, however, that y is observed if y* > 0 and is not observed ify* < 0. Then the observed y, will be defined as

y* = x. + Ui if y* > 0

[0 ifyJO (3 19)

ui ~ IN(0, tr)

This is known as the tobit model (Tobins probit) and was first analyzed in the econometrics literature by Tobin." It is also known as a censored normal regression model because some observations on y* (those for which y* < 0) are censored (we are not allowed to see them). Our objective is to estimate the parameters p and a.

Some Examples

The example that Tobin considered was that of automobile expenditures. Suppose that we have data on a sample of households. We wish to estimate, say, the income elasticity of demand for automobiles. Let y* denote expenditures on automobiles and x denote income, and we postulate the regression equation

y; = px, + , ui ~ IN(0, cT)

However, in the sample we would have a large number of observations for which the expenditures on automobiles is zero. Tobin argued that we should use the censored regression model. We can specify the model as

for those with positive automobile expenditures ,„ for those with no expenditures

The structure of this model thus appears to be the same as that in (8.19).

There have been a very large number of applications of the tobit model." Take, for instance, hours worked (H) or wages {W). If we have observations on a number of individuals, some of whom are employed and others not, we can specify the model for hours worked as

„ fpx, + u,

for those working

for those who are not working

Similarly, for wages we can specify the model

yZi + v( for those working

0 for those who are not working

(8.21)

(8.22)

J. Tobin, "Estimation of Relationships for Limited Dependent Variables," Econometrica, Vol. 26, 1958, pp. 24-36.

See T. Amemiya, "Tobit Models: A Survey," Journal of Econometrics, Vol. 24, 1984, pp. 3-61, which lists a large number of applications of the tobit model.

The structure of these models again appears to be the same as in (8.19). However, there are some limitations in the formulation of the models in (8.20)-(8.22) that we will presently outline after discussing the estimation of the model in (8.19).

Method of Estimation

Let us consider the estimation of p and a by the use of ordinary least squares (OLS). We cannot use OLS using the positive observations y, because when we write the model

y, = px, -I- u,

the error term u, does not have a zero mean. Since observations with y* < 0 are omitted, it implies that only observations for which u,> - px, are included in the sample. Thus the distribution of u, is a truncated normal distribution shown in Figure 8.4 and its mean is not zero. In fact, it depends on p, a, and x, and is thus different for each observation. A method of estimation commonly suggested is the maximum likelihood (ML) method, which is as follows: Note that we have two sets of observations:

1. The positive values of y, for which we can write down the normal density function as usual. We note that (y, - ,)/ has a standard normal distribution. ,

2. The zero observations of for which all we know is that y* < 0 or px, + u, < 0. Since uja has a standard normal distribution, we will write this as uJa < -(Px,)/ct. The probability of this can be written as F(-(Px,)/ct), where F{z) is the cumulative distribution function of the standard normal.

Let us denote the density function of the standard normal by /(•) and the cumulative distribution function by F(). Thus

1 / tA

fit) = 77 exp

-(ic/a

Figure 8.4. Truncated normal distribution.